On the semi-simplicity of the U_{p}-operator on modular forms.

Robert F. Coleman  Bas Edixhoven^{∗}

Fall 1996

1 Introduction.

For N and k positive integers, let M^{0}(N, k)_{ℂ} denote the ℂ-vector space of cuspidal modular forms of level N and weight k. This vector space is equipped with the usual Hecke operators T_{n}, n ≥ 1. If we need to consider several levels or weights at the same time, we will denote this T_{n} by T_{n}^{N}, or T_{n}^{N,k}. If p is a prime number dividing N, our T_{p} is also known under the name U_{p}. One of our main results can be stated very easily: if k = 2 and p^{3} does not divide N, then the operator T_{p} is semi-simple. We can prove the same result for weight k ≥ 3, under the assumption that certain crystalline Frobenius elements are semi-simple. Milne has shown in [11, §2] that this semi-simplicity is implied by Tate’s conjecture claiming that for X projective and smooth over a finite field of characteristic p, and r ≥ 0, dim_{ℚ}(CH^{r}(X)/≡_{num}) equals the order of ζ(X, s) at r. Ulmer proved in [17] that T_{p} is semi-simple, for k = 3 and p^{2} not dividing N, under the assumption of the Birch-Swinnerton-Dyer conjecture for elliptic curves over function fields in characteristic p. His method is quite different from ours: assuming that T_{p} is not semi-simple, he really shows that the Birch-Swinnerton-Dyer conjecture does not hold for an explicitly given elliptic curve.

The structure of our proof is as follows. Using the theory of newforms, the problem is shown to be equivalent to the problem of showing that, for a normalized newform f of weight k, prime-to-p level and character ε, the polynomial x^{2} − a_{p}x + ε(p)p^{k−1} has no double root. This polynomial happens to be the characteristic polynomial of the Frobenius element at p in the two-dimensional Galois representations associated to f; it is also the characteristic polynomial of the crystalline Frobenius asociated to f. We show that this crystalline Frobenius cannot be a scalar.

In Sections 2 and 3 we prove the results concerning these Frobenius elements for k = 2 and k ≥ 2, respectively. Section 2 is quite elementary, whereas in Section 3 we use a lot of the machinary for comparing p-adic etale and crystalline cohomology. In Section 4 we give some applications: the Ramanujan inequality is a strict inequality in certain cases, certain Hecke algebras are reduced, hence have non-zero discriminant. Section 5 gives some results, due to Abbes and Ullmo, concerning the discriminants of certain Hecke algebras.

∗partially supported by the Institut Universitaire de France

To end this introduction, let us explain why the case k = 1 is completely different. Consider a normalized cuspidal eigenform f = ∑ a_{n}q^{n} of some level N, of weight one and with some character ε. Deligne and Serre have shown ([5, §4]) that there exists a continuous representation ρ_{f} from Gal(\overline{ℚ}/ℚ) to GL_{2}(ℂ), unramified outside N, such that, for all primes p not dividing N, the characteristic polynomial of a Frobenius element at p is x^{2} − a_{p}x + ε(p). Since the image of ρ_{f} is finite, Chebotarev’s density theorem gives the existence of primes p not dividing N such that the Frobenius element at p is the identity element, hence has characteristic polynomial (x − 1)^{2}.

2 An elementary proof in the case of weight two.

2.1 Theorem. Let f = ∑ a_{n}q^{n} be a cuspidal normalized eigenform of weight two, some level N and character ε: (ℤ/Nℤ)^{∗} → ℂ^{∗}. Let p be a prime number not dividing N. Then the polynomial x^{2} − a_{p}x + ε(p)p has simple roots.

Proof. The proof is by contradiction, so we suppose that the polynomial has a double root λ. Then of course we have λ^{2} = ε(p)p and 2λ = a_{p}. Let K be the finite extension of ℚ generated by the a_{n} and the ε(a), and let O_{K} be its ring of integers. Let J denote the jacobian variety of the modular curve X_{1}(N) over ℚ. We identify the space M^{0}(N, 2)_{ℚ} of weight two cuspforms of level N and with coefficients in ℚ with the cotangent space at the origin of J; this is compatible with the action of the Hecke operators. Let 𝕋 be the subring of End(J) that is generated by the T_{n}, n ≥ 1, and the diamond operators ⟨a⟩, a ∈ (ℤ/Nℤ)^{∗}. Let I be the annihilator of f in 𝕋, and let A′_{ℚ} := J/IJ be the quotient of J by its subvariety generated by the images of all elements of I. It is well known that A′_{ℚ} has dimension [K : ℚ], and that for every prime number l, the free K ⊗ ℚ_{l}-module V_{l}(A′_{ℚ}) of rank two gives the l-adic Galois representation ρ_{f,l} associated to f. We prefer to work with an abelian variety A_{ℚ} that is isogeneous to A′_{ℚ} and on which we have an action of all of O_{K}. This is easily done: define A_{ℚ} := O_{K} ⊗_{𝕋} A′_{ℚ}, where the tensor product should be calculated by taking a presentation of O_{K}.

The abelian variety A_{ℚ} has good reduction at p; let A_{ℤ_{p}} denote the corresponding abelian scheme over ℤ_{p}. We consider the first algebraic de Rham cohomology group M := H_{DR}^{1}(A_{ℤ_{p}}/ℤ_{p}). It is a free ℤ_{p}-module of rank 2[K : ℚ], equipped with its Hodge filtration:

(2.2)  M = Fil^{0}M ⊃ Fil^{1}M = H^{0}(A_{ℤ_{p}}, Ω^{1}).

The submodule Fil^{1}M is free of rank [K : ℚ] as ℤ_{p}-module, and has the property that Fil^{0}M/Fil^{1}M is torsion free. The double root λ of x^{2} − a_{p}x + ε(p)p is in O_{K}, since it is integral and 2λ is in K. In the endomorphism ring of A_{𝔽_{p}} we have the Eichler-Shimura congruence relation:

(2.3)  0 = (Frob_{p} − Frob_{p})(Frob_{p} − Frob′_{p}) = Frob_{p}^{2} − a_{p}Frob_{p} + ε(p)p = (Frob_{p} − λ)^{2},

where Frob_{p} is the Frobenius endomorphism and Frob′_{p} the Verschiebung, multiplied by ε(p). The fact that every abelian variety over 𝔽_{p} is isogeneous to a product of simple ones implies that Frob_{p} is semi-simple in the sense that it satisfies an identity of the form P (Frob_{p}) = 0, with P a polynomial with coefficients in ℚ that has simple roots. It follows that Frob_{p} = λ in End(A_{𝔽_{p}}). Since O_{K} ⊗ ℤ_{p} is a product of a finite number of discrete valuation rings, Fil^{1}M is a locally free module over it; it is in fact free of rank one (consider ℚ ⊗ Fil^{1}M). It follows that λ does not annihilate 𝔽_{p} ⊗ Fil^{1}M, since we have λ^{2} = ε(p)p. But 𝔽_{p} ⊗ Fil^{1}M is the same as H^{0}(A_{𝔽_{p}}, Ω^{1}), and on this module λ acts as Frob_{p}^{∗}, hence it does annihilate. This contradiction finishes the proof. □

3 The general case.

In this section we try to generalize Theorem 2.1 as much as we can to higher weights. For doing that we replace the module M of Section 2 by the p-adic crystalline realization of the motive associated to f; this gives us a filtered ϕ-module M of rank two. The comparison theorem for crystalline and p-adic etale cohomology implies that this filtered ϕ-module is weakly admissible, from which it follows immediately that the crystalline Frobenius ϕ cannot be a scalar. Unfortunately, it is not known that ϕ is semi-simple, so all we show is that semi-simplicity of ϕ implies that the polynomial x^{2} − a_{p}x + ε(p)p^{k−1} has simple roots.

Let f = ∑ a_{n}q^{n} be a normalized cuspidal newform of some level N, weight k ≥ 2 and character ε. Let K be the field generated by the a_{n} and the ε(a). Let p be a prime number not dividing N. Our first objective is to construct the p-adic crystalline realization of the motive associated to f. In [14], Scholl constructs a Grothendieck motive M(f) over ℚ, with coefficients in K, such that for every prime number l the Galois representation ρ_{f,l}: G_{ℚ} → GL_{2}(ℚ_{l} ⊗ K) is the dual of the l-adic realization H_{l}(M(f)). Concretely, he constructs a projector in the group ring of a finite group of automorphisms of the smooth and projective model X of the k−2-fold fibered product of the universal elliptic curve over Y (N′) (with N′ a suitable multiple of N) constructed by Deligne in [3, §5], such that the l-adic and Betti realizations of the corresponding Chow motive are, in a way that is compatible with Hecke operators, the parabolic cohomology groups used in [3]. The Grothendieck motive associated to this Chow motive (i.e., one replaces rational equivalence by homological equivalence) has an action by the Hecke algebra of the space of cuspidal modular forms of weight k on the modular curve X(N′)_{ℚ}; M(f) is a suitable factor. The variety X over ℚ has a smooth projective model over ℤ[1/N′] (see [14, 4.2.1] and [3]), hence M(f) has a crystalline realization M := H_{crys}(M(f)) which is a free ℚ_{p} ⊗ K-module of rank two equipped with an endomorphism ϕ, the crystalline Frobenius, that is induced by the Frobenius endomorphism of the reduction mod p of X. The characteristic polynomial of ϕ is x^{2} − a_{p}x + ε(p)p^{k−1}; this can be shown in the same way as one can show it for the l-adic realizations, or one invokes a result of Katz and Messing (see [14, 4.2.3]).

3.1 Theorem. Let f = ∑ a_{n}q^{n} be a normalized cuspidal newform of some level N, weight k ≥ 2 and character ε. Let p be a prime number not dividing N. Then the Frobenius ϕ of the crystalline realization M of the motive M(f) is not scalar, i.e., it is not in ℚ_{p} ⊗ K.

Proof. The proof is by contradiction. We suppose that ϕ is an element, λ say, of ℚ_{p} ⊗ K. The comparison theorem for crystalline and de Rham cohomology for smooth proper ℤ_{p}-schemes (see [8, §1.3]) gives us an isomorphism between M and ℚ_{p} ⊗ H_{DR}(M(f)), and hence a decreasing filtration (the Hodge filtration) Fil on M. So (M, ϕ, Fil) is an object of the category of filtered ϕ-modules. (A filtered ϕ-module is a finite dimensional ℚ_{p}-vector space M with a decreasing, exhaustive and separating filtration Fil^{i}, i ∈ ℤ, and an endomorphism ϕ; morphisms are linear maps respecting Fil and ϕ; see [8, §2.3].) The Hodge filtration on H_{DR}(M(f)) induces the Hodge decomposition of ℂ ⊗ H_{B}(M(f)), which is of type (k−1, 0), (0, k−1), hence Fil^{0}(M) = M, Fil^{1}(M) = Fil^{k−1}(M) is free of rank one, and Fil^{k}(M) = 0. Fontaine has constructed Grothendieck’s “mysterious functor” D_{crys} from the category of finite dimension ℚ_{p}-vector spaces with continuous Gal(\overline{ℚ}_{p}/ℚ_{p})-action to the category of filtered ϕ-modules (see the introduction of [8]). It is a theorem of Faltings (see [8, §3.2]), of which a special case was proved earlier by Fontaine and Messing, that there is an isomorphism of filtered ϕ-modules between M and D_{crys}(H_{p}(M(f))). In fact, the theorem is stated for varieties, but since the isomorphism is compatible with the multiplicative structure and with cycle classes, it also works for Grothendieck motives.

The most important consequence of this theorem for us is that the filtered ϕ-module M is admissible, hence weakly admissible, in the sense of Fontaine, see [7, 4.4.6]. Recall that to a filtered ϕ-module M one associates two polygons: the Hodge polygon, depending only on the filtration, and the Newton polygon, depending only on ϕ. Weakly admissible means that for every subobject M′ of M the Newton polygon lies above or on the Hodge polygon, and that the two polygons for M itself have the same end-point. An equivalent formulation is the following. For M a filtered ϕ-module let t_{N}(M) be the p-adic valuation of the determinant of ϕ, and let t_{H}(M) be the maximal i such that Fil^{i}(det M) ≠ 0. Then M is weakly admissible if and only if firstly: t_{N}(M) = t_{H}(M), and secondly: for all subobjects M′ of M one has t_{H}(M′) ≤ t_{N}(M′).

Consider now our weakly admissible filtered ϕ-module M. Since ϕ is the element λ of ℚ_{p} ⊗ K, we have the subobject M′ := Fil^{k−1}(M) of M (we give it the induced filtration). Then t_{H}(M′) = [K : ℚ](k − 1), whereas t_{N}(M′) = [K : ℚ](k − 1)/2 (recall that M′ is free of rank one over ℚ_{p} ⊗ K and that λ^{2} = ε(p)p^{k−1}). Since k ≥ 2, this contradicts the weak admissibility of M. □

3.2 Corollary. Let f = ∑ a_{n}q^{n} be a normalized cuspidal eigenform of some level N, weight k ≥ 2 and character ε. Let p be a prime number not dividing N and suppose that the crystalline Frobenius ϕ of the ℚ_{p}-vector space M(f) is semi-simple. Then the polynomial x^{2} − a_{p}x + ε(p)p^{k−1} has simple roots.

Proof. The proof is by contradiction: we suppose that λ ∈ K is a double root. As we have already said above, the polynomial in question is the characteristic polynomial of the endomorphism ϕ of the free rank two ℚ_{p} ⊗ K-module M. Hence it satisfies the identity (ϕ − λ)^{2} = 0. Now ℚ_{p} ⊗ K is a finite product of fields, hence the semi-simplicity of ϕ implies that ϕ is multiplication by λ. But this contradicts Theorem 3.1. □

3.3 Remark. The first three lines of [11, §2] show that Tate’s conjecture mentioned in Section 1 implies the semi-simplicity of l-adic and crystalline Frobenius elements of smooth projective varieties over finite fields. □

3.4 Remark. Scholl remarks that his explicit construction of the crystalline realization M of M(f) in [15] should show directly that M is weakly admissible. □

4 Applications.

4.1 Theorem. Let N ≥ 1 and k ≥ 2 be integers. Let f = ∑ a_{n}q^{n} be a normalized cuspidal eigenform of level N and weight k. Let p be a prime number not dividing N. If k > 2 assume Tate’s conjecture mentioned in Section 1. Then we have |a_{p}| < 2p^{(k−1)/2}.

Proof. Let ε be the character of f. Theorems 2.1 and 3.1 show that x^{2} − a_{p}x + ε(p)p^{k−1} has no double root. Deligne has shown ([5, Thm. 6.1] and [4, Thm. 1.6]) that the roots have absolute value p^{(k−1)/2}. □

4.2 Theorem. Let N ≥ 1 and k ≥ 2 be integers. Let p be a prime number such that p^{3} does not divide N. Assume Tate’s conjecture mentioned in Section 1 if k ≥ 3 and p|N. Then the endomorphism T_{p} of M^{0}(N, k)_{ℂ} is semi-simple.

Proof. For the sake of notation, let N ≥ 1 be an integer and p a prime number not dividing N. Let k ≥ 1. In this case T_{p} is normal with respect to the Petersson scalar product on M^{0}(N, k)_{ℂ}. Hence T_{p} is diagonalizable.

Let us now consider M^{0}(pN, k)_{ℂ}, with k ≥ 2. By the theory of newforms, M^{0}(pN, k)_{ℂ} is the direct sum of its p-new part M^{0}(pN, k)_{ℂ}^{pnew} and its p-old part M^{0}(pN, k)_{ℂ}^{pold}, this decomposition being respected by all Hecke operators. The restriction of T_{p}^{pN,k} to M^{0}(pN, k)_{ℂ}^{pnew} is normal, hence diagonalizable. The p-old part is isomorphic to the direct sum of two copies of M^{0}(N, k)_{ℂ}, via the map (f(q), g(q)) ↦ f(q) + g(q^{p}). The restriction of T_{p}^{pN,k} to M^{0}(pN, k)_{ℂ}^{pold} is then given by the following two by two matrix:

(4.2.1)  T_{p}^{pN,k}|_{M^{0}(pN,k)_{ℂ}^{pold}} = ( \begin{matrix}T_{p}^{N,k} & 1 \\ −p^{k−1}⟨p⟩ & 0 \end{matrix} ) .

We have already seen that M^{0}(N, k)_{ℂ} is the direct sum of its common eigenspaces V_{a_{p},ε} for T_{p} and the diamond operators. It follows that M^{0}(pN, k)_{ℂ}^{pold} decomposes as a direct sum of terms V_{a_{p},ε}^{2}, and that the restriction of T_{p}^{pN,k} to each of the V_{a_{p},ε}^{2} is annihilated by x^{2} − a_{p}x + ε(p)p^{k−1}. Under the hypotheses of the theorem we are proving, these polynomials have simple roots by Theorems 2.1 and 3.1.

Let us now consider M^{0}(p^{2}N, k)_{ℂ}, with k ≥ 2. Here too this space is the direct sum of its p-old and p-new parts. On the p-new part T_{p}^{p^{2}N,k} is self-adjoint, hence diagonalizable. The p-old part is now isomorphic to the direct sum of three copies of M^{0}(N, k)_{ℂ} and two copies of M^{0}(pN, k)_{ℂ}^{pnew}. The restrictions of T_{p}^{p^{2}N,k} to M^{0}(N, k)_{ℂ}^{3} and (M^{0}(pN, k)_{ℂ}^{pnew})^{2} are given by the following matrices:

(4.2.2)  ( \begin{matrix}T_{p}^{N,k} & 1 & 0 \\ −p^{k−1}⟨p⟩ & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) ,  ( \begin{matrix}T_{p}^{pN,k} & 1 \\ 0 & 0 \end{matrix} )

One can now repeat the same type of argument as above, invoking Theorems 2.1 and 3.1 to see that, under the hypotheses of the theorem we are proving, x(x^{2} − a_{p}x + ε(p)p^{k−1}), with (a_{p}, ε(p)) as before, has simple roots. The space M^{0}(pN, k)_{ℂ}^{pnew} is a direct sum of eigenspaces for T_{p}^{Np,k}, and one knows that the eigenvalues are non-zero (see [5, §1.8]). It follows that also the restriction of T_{p}^{p^{2}N,k} to (M^{0}(pN, k)_{ℂ}^{pnew})^{2} is diagonalizable. □

4.3 Corollary. Let N ≥ 1 be cube free, and let k ≥ 2. Let 𝕋 be the ℤ-algebra generated by the endomorphisms T_{n}, n ≥ 1, and ⟨a⟩, a ∈ (ℤ/Nℤ)^{∗}, of M^{0}(N, k)_{ℂ}. Assume Tate’s conjecture mentioned in Section 1 if k > 2. Then the ring 𝕋 is reduced.

Proof. This is so because 𝕋 is a subring of the ℂ-algebra 𝕋_{ℂ} := ℂ ⊗ 𝕋 generated by the T_{n} and ⟨a⟩. Theorem 4.2 tells us that the T_{p} and ⟨a⟩ can be simultaneously diagonalized. They generate 𝕋_{ℂ}, hence 𝕋_{ℂ} is a product of copies of ℂ, hence reduced. □

4.4 Remark. For general N and k, the Hecke algebra 𝕋 is well known to be a free ℤ-module; 𝕋_{ℚ} := ℚ ⊗ 𝕋 is well known to be Gorenstein, i.e., its ℚ-linear dual (ℚ ⊗ 𝕋)^{∨} is free of rank one as ℚ ⊗ 𝕋-module, see for example [18, p. 481]. One way to prove this is as follows. By the q-expansion principle, M^{0}(N, k)_{ℂ}^{∨} is free of rank one as 𝕋_{ℂ}-module. Then one constructs a 𝕋_{ℂ}-bilinear ℂ-valued pairing on M^{0}(N, k)_{ℂ} to get an isomorphism of 𝕋_{ℂ}-modules between M^{0}(N, k)_{ℂ} and its dual. Another way to prove it is to use the theory of new forms. This last proof gives more information on how exactly 𝕋_{ℂ} decomposes as a product of ℂ-algebras. Parent needed such information and so he worked out the details in [13]. His work made see that Theorem 4.2 should be stated for N cube free instead of square free. Of course, statements that completions of 𝕋 at certain of its maximal ideals are Gorenstein are much more subtle and harder to prove (see for example [18, §2.1]). □

5 Discriminants of Hecke algebras.

According to Corollary 4.3, certain Hecke algebras 𝕋 are reduced. This means that the discriminants discr(𝕋) of their trace forms (x, y) ↦ trace(xy) are non-zero. These discriminants “count” all congruences between different eigenforms of fixed level and weight, hence are quite useless for dealing with congruences with a fixed form (in particular, nothing interesting can be said on the degrees of modular parametrizations of elliptic curves over ℚ). The following result, relating such discriminants to heights of modular curves, is due to Abbes and Ullmo (unpublished).

5.1 Theorem. (Abbes, Ullmo) Let p be a prime number, and let 𝕋 be the Hecke algebra associated to M^{0}(p, 2)_{ℂ} = H^{0}(X_{0}(p)_{ℂ}, Ω). Then one has:

h(X_{0}(p)_{ℚ}) = \frac{1}{2} log |discr(𝕋)| − ∑_{i=1}^{g} log ∥ω_{i}∥,

where h is the modular height of curves over ℚ (see [16, §3.3]), where ω_{1}, …, ω_{g} are the normalized eigenforms and ∥ · ∥ the norm of the scalar product ⟨ω|η⟩ = (i/2) ∫_{X_{0}(p)(ℂ)} ω ∧ \overline{η} on H^{0}(X_{0}(p)_{ℂ}, Ω).

Proof. We start by recalling the definition of h. So let X_{ℚ} be a smooth proper geometrically irreducible curve over ℚ, of some genus g. Let J_{ℚ} be its jacobian and J the Néron model over ℤ. Then we have the free ℤ-module of rank one ω_{J} := Λ^{g}0^{∗}Ω_{J/ℤ}^{1}, with the scalar product on ℂ ⊗ ω_{J} given by ⟨ω|η⟩ = (i/2)^{g}(−1)^{g(g−1)/2} ∫_{J(ℂ)} ω ∧ \overline{η}. The height h(X_{ℚ}) is then defined to be the Arakelov degree of this metrized line bundle: h(X_{ℚ}) = deg_{Ar}(ω_{J}) = − log ∥ω∥, with ω a generator of ω_{J}. The ℤ-module ω_{J} is equipped with the scalar product on ℂ ⊗ ω_{J} = H^{0}(X_{0}(p)_{ℂ}, Ω) already mentioned in the theorem above. This scalar product induces a real scalar product, and hence a volume form, on ℝ ⊗ ω_{J} (the volume form being determined by the condition that a cube with edges of length one has volume one). A calculation (see [16, lemme 3.2.1]) shows that one has:

(5.1.1)  h(X_{ℚ}) = − log vol(ℝ ⊗ ω_{J}/ω_{J}).

Let now X be the curve X_{0}(p)_{ℚ}. In that case, ω_{J} is the same as H^{0}(X_{ℤ}, Ω), with X_{ℤ} the usual model over ℤ (semi-stable, X_{𝔽_{p}} consisting of two irreducible components) and Ω its dualizing sheaf ([10, Ch. II, §3]). The fact that the two irreducible components of X_{𝔽_{p}} are of genus zero implies that the pairing

(5.1.2)  𝕋 × ω_{J} → ℤ, (t, ω) ↦ a_{1}(tω),

with a_{1} denoting the linear form on ω_{J} that takes the coefficient of q in the q-expansion, is perfect, i.e., it induces an isomorphism between 𝕋^{∨} of 𝕋 and ω_{J}. Let ω_{1}, …, ω_{g} be as in the theorem. Sending an element of 𝕋_{ℝ} to the eigenvalues of the ω_{i} for it is an isomorphism of ℝ-algebras:

(5.1.3)  𝕋_{ℝ} → ℝ^{g}, t ↦ (a_{1}(tω_{1}), …, a_{1}(tω_{g})).

The trace form on 𝕋 corresponds to the standard scalar product on ℝ^{g}. Composing the dual of the isomorphism (5.1.3) with the isomorphism 𝕋_{ℝ}^{∨} → ℝ⊗ω_{J} from (5.1.2) gives an isomorphism from ℝ^{g} to ℝ ⊗ ω_{J} mapping the ith standard basis vector e_{i} to ω_{i}. It follows that the volume form on ℝ ⊗ ω_{J} corresponds to ∏_{i} ∥ω_{i}∥ times the one on 𝕋_{ℝ}^{∨} corresponding to the trace form. We find:

(5.1.4)  vol(ℝ ⊗ ω_{J}/ω_{J}) = (∏_{i=1}^{g} ∥ω_{i}∥) vol(𝕋_{ℝ}^{∨}/𝕋) = (∏_{i=1}^{g} ∥ω_{i}∥) vol(𝕋_{ℝ}/𝕋)^{−1}.

The proof is finished since vol(𝕋_{ℝ}/𝕋) = |discr(𝕋)|^{1/2}. □

5.2 Theorem. (Abbes, Ullmo) For every ε > 0 there exists c(ε) in ℝ such that for all prime numbers p one has h(X_{0}(p)_{ℚ}) ≤ c(ε)p^{1+ε}.

Proof. Let 𝕋 and g be as above, and let 𝕋′ := ∑_{i=1}^{g} ℤT_{i}. Then 𝕋′ is of finite index in 𝕋 because ∞ is not a Weierstrass point of X_{0}(p) (see [9, §3] or [6, §4]). The image of T_{i} under the isomorphism (5.1.3) is (a_{i}(ω_{1}), …, a_{i}(ω_{g})). It follows that we have the equalities:

(5.2.1)  discr(𝕋) = discr(𝕋′)|𝕋/𝕋′|^{−2}, |discr(𝕋′)| = | det_{1≤i,j≤g} a_{i}(ω_{j})|^{2}.

Weil’s theorem on absolute values of eigenvalues of Frobenius endomorphisms of abelian varieties over finite fields implies that |a_{i}(ω_{j})| ≤ σ(i)i^{1/2}, where σ(i) is the number of positive integers dividing i. It follows that |discr(𝕋′)| ≤ g! ∏_{i=1}^{g} σ(i)i^{1/2}. The rest of the proof consists of applying Theorem 5.1 and standard estimates (including an absolute lower bound for the ∥ω_{i}∥). □

5.3 Remark. One knows (see [10, Ch. II, Prop. 10.6]) that Spec(𝕋) (for 𝕋 as above) is connected. This implies a lower bound for discr(𝕋) (use [12]). On the other hand, the ∥ω_{i}∥ are bounded above by a constant times p. Unfortunately, the lower bound for h(X_{0}(p)_{ℚ}) obtained like this seems too weak to be useful. Assuming 𝕋 to be Gorenstein does not significantly improve this lower bound. □

5.4 Remark. Several problems arise when one wants to generalize the above results for X_{0}(p)_{ℚ} to more general X_{0}(N)_{ℚ}. First of all, 𝕋^{∨} will not be the same as ω_{J}, but it should be possible to estimate |𝕋^{∨}/ω_{J}|. Secondly, the comparison of the trace form on 𝕋 and the scalar product on ℝ ⊗ ω_{J} is more complicated. Note that the trace form can even be degenerate, if N is not cube free. Thirdly, it seems to be unknown if ∞ can be a Weierstrass point on X_{0}(N)_{ℚ} when N is square free (see [9, §3]; it is known that ∞ is a Weierstrass point when N is divisible by 4 or 9, for example).

Suppose now that N is square free. Before knowing the result that 𝕋 is reduced, Abbes and Ullmo have related h(X_{0}(N)) to the discriminants of the “new parts” of the Hecke algebras of level dividing N (unpublished). The techniques they use come from [1] and [2]. They show that Theorem 5.2 holds for square free N such that ∞ is not a Weierstrass point. With the same techniques it is certainly possible to solve the first two of the three problems mentioned above. □

Acknowledgements. We thank Rutger Noot for a discussion during which we were led to a proof of Theorem 2.1. Bas Edixhoven is grateful to Richard Taylor for a discussion concerning the case of general weights. He also thanks Pierre Berthelot for explaining to him certain results on Newton and Hodge polygons. Thanks also go to Ahmed Abbes and Emmanuel Ullmo for communicating to us their results mentioned in Section 5, and permitting us to publish them. Bas Edixhoven wants to thank the Centre for Research in Mathematics at the Institut d’Estudis Catalans in Barcelona where he finally found time to write this text. Robert Coleman thanks the University of Rennes for inviting him for a month in 1995.

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Robert Coleman Dept. of Mathematics University of California Berkeley, CA 94720 U.S.A.

Bas Edixhoven

IRMAR

Campus de Beaulieu 35042 Rennes cedex France